Question

Prove that is irrational.

Answer 1

Let is a rational number.

Therefore, we can find two integers a, b (b ≠ 0) such that

Let a and b have a common factor other than 1. Then we can divide them by the common factor, and assume that a and b are co-prime.

Therefore, a² is divisible by 5 and it can be said that a is divisible by 5.

Let a = 5k, where k is an integer

This means that b² is divisible by 5 and hence, b is divisible by 5.

This implies that a and b have 5 as a common factor.

And this is a contradiction to the fact that a and b are co-prime.

Hence,cannot be expressed as or it can be said that is irrational.